Average norms of polynomials

AVERAGE MAHLER’S MEASURE AND Lp NORMS OF UNIMODULAR POLYNOMIALS

A polynomial f ∈ C[z] is unimodular if all its coefficients have unit modulus. Let Un denote the set of unimodular polynomials of degree n−1, and let Un denote the subset of reciprocal unimodular polynomials, which have the property that f(z) = ωzn−1f(1/z) for some complex number ω with |ω| = 1. We study the geometric and arithmetic mean values of both the normalized Mahler’s measure M(f)/ √ n ...

AVERAGE MAHLER’S MEASURE AND Lp NORMS OF LITTLEWOOD POLYNOMIALS

Littlewood polynomials are polynomials with each of their coefficients in the set {−1, 1}. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1. We show that the arithmetic means of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1 are asymptotically e−γ/2 √ n and Γ(1+ p/2)1...

Norms of Factors of Polynomials

for any positive integer k is bounded above by a quantity that is independent of k. Hence, whenever A(x) is divisible by a cylcotomic polynomial and N is su ciently large, there will be Q(x) 2 Z[x] with arbitrarily large Euclidean norm and with kAQk N . It is reasonable, however, to expect that the Euclidean norm of Q(x) is bounded whenever A(x) is free of cyclotomic factors. This in fact is th...

L norms of trigonometric polynomials

On discrete norms of polynomials

For a polynomial p of degree n<N we compare two norms: ‖p‖ := sup{|p(z)| : z ∈ C; |z| = 1} and ‖p‖N := sup {∣∣p (zj )∣∣ : j = 0, . . . , N − 1} ; zj = e2 i j N . We show that there exist universal constants C1 and C2 such that 1+ C1 log ( N N − n ) sup { ‖p‖ ‖p‖N : p ∈ Pn } C2 log ( N N − n ) + 1. © 2005 Elsevier Inc. All rights reserved.